complexOTHEReigen(3)

📦

version 3.11.0 · 2 versions

3.11.0 · debian bookworm (latest)
3.10.0 · ubuntu jammy

✨ complexOTHEReigen(3) ✨

okay so bestie, welcome to the LAPACK glow-up corner 🫶 this is the manual page for complexOTHEReigen and it is long, so we're doing it in ten parts. grab a snack, hydrate, we're going in 💅

🌸 NAME 🌸

complexOTHEReigen — complex Other Eigenvalue routines ✨

🦋 SYNOPSIS 🦋

🌈 Functions 🌈

subroutine cggglm (N, M, P, A, LDA, B, LDB, D, X, Y, WORK, LWORK, INFO)
subroutine chbev (JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, RWORK, INFO)
subroutine chbev_2stage (JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, LWORK, RWORK, INFO)
subroutine chbevd (JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO)
subroutine chbevd_2stage (JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO)
subroutine chbevx (JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK, IWORK, IFAIL, INFO)
subroutine chbevx_2stage (JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, LWORK, RWORK, IWORK, IFAIL, INFO)
subroutine chbgv (JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, Z, LDZ, WORK, RWORK, INFO)
subroutine chbgvd (JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, Z, LDZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO)
subroutine chbgvx (JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK, IWORK, IFAIL, INFO)
subroutine chpev (JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, RWORK, INFO)
subroutine chpevd (JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO)
subroutine chpevx (JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK, IWORK, IFAIL, INFO)
subroutine chpgv (ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ, WORK, RWORK, INFO)
subroutine chpgvd (ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO)
subroutine chpgvx (ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK, IWORK, IFAIL, INFO)

cggglm — CGGGLM, the general Gauss-Markov linear model solver 💜
chbev — CHBEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices 🌸
chbev_2stage — CHBEV_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices ✨
chbevd — CHBEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices 💫
chbevd_2stage — CHBEVD_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices 🎀
chbevx — CHBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices 🦋
chbevx_2stage — CHBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices ⭐
chbgv — CHBGV 💖
chbgvd — CHBGVD 💖
chbgvx — CHBGVX 💖
chpev — CHPEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices 🌷
chpevd — CHPEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices 🌺
chpevx — CHPEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices 🧚
chpgv — CHPGV 💜
chpgvd — CHPGVD 💜
chpgvx — CHPGVX 💜

🔥 Detailed Description 🔥

This is the group of complex Other Eigenvalue routines. that's it, that's the vibe, no cap 💅

👑 Function Documentation 👑

💖 subroutine cggglm 💖

subroutine cggglm (integer N, integer M, integer P, complex, dimension( lda,
    * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB,
    complex, dimension( * ) D, complex, dimension( * ) X, complex,
    dimension( * ) Y, complex, dimension( * ) WORK, integer LWORK, integer
    INFO)

CGGGLM 🫶

🌟 Purpose 🌟

CGGGLM solves a general Gauss-Markov linear model (GLM) problem, and bestie it is so specific about what that means:

     minimize || y ||_2   subject to   d = A*x + B*y
         x

so like, A is an N-by-M matrix, B is an N-by-P matrix, and d is a given N-vector. the assumptions here are that M <= N <= M+P, and:

            rank(A) = M    and    rank( A B ) = N.

under those assumptions, the constrained equation is always consistent (iconic), and there is a unique solution x and a minimal 2-norm solution y. those come out of a generalized QR factorization of the matrices (A, B), which goes like:

    A = Q*(R),   B = Q*T*Z.
          (0)

and here's the cute special case: if matrix B is square nonsingular, then the GLM problem is equivalent to this weighted linear least squares problem:

              minimize || inv(B)*(d-A*x) ||_2
                  x

where inv(B) means the inverse of B. no cap 💅

🎀 Parameters 🎀

N — N is INTEGER. the number of rows of the matrices A and B. N >= 0.
M — M is INTEGER. the number of columns of the matrix A. 0 <= M <= N.
P — P is INTEGER. the number of columns of the matrix B. P >= N-M.
A — A is COMPLEX array, dimension (LDA,M).
- on entry, the N-by-M matrix A.
- on exit, the upper triangular part of the array A contains the M-by-M upper triangular matrix R.
LDA — LDA is INTEGER. the leading dimension of the array A. LDA >= max(1,N).
B — B is COMPLEX array, dimension (LDB,P).
- on entry, the N-by-P matrix B.
- on exit, if N <= P, the upper triangle of the subarray B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T; if N > P, the elements on and above the (N-P)th subdiagonal contain the N-by-P upper trapezoidal matrix T.
LDB — LDB is INTEGER. the leading dimension of the array B. LDB >= max(1,N).
D — D is COMPLEX array, dimension (N).
- on entry, D is the left hand side of the GLM equation.
- on exit, D is destroyed. 💀 gone. that's the deal, sorry bestie.
X — X is COMPLEX array, dimension (M).
Y — Y is COMPLEX array, dimension (P).
- on exit, X and Y are the solutions of the GLM problem. 🎉
WORK — WORK is COMPLEX array, dimension (MAX(1,LWORK)).
- on exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK — LWORK is INTEGER. the dimension of the array WORK. LWORK >= max(1,N+M+P). for optimum performance, LWORK >= M+min(N,P)+max(N,P)*NB, where NB is an upper bound for the optimal blocksizes for CGEQRF, CGERQF, CUNMQR and CUNMRQ.

if LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. ✨

INFO — INFO is INTEGER.
- = 0: successful exit. 🥳
- < 0: if INFO = -i, the i-th argument had an illegal value. 🙈
- = 1: the upper triangular factor R associated with A in the generalized QR factorization of the pair (A, B) is singular, so that rank(A) < M; the least squares solution could not be computed. 😭
- = 2: the bottom (N-M) by (N-M) part of the upper trapezoidal factor T associated with B in the generalized QR factorization of the pair (A, B) is singular, so that rank( A B ) < N; the least squares solution could not be computed. 😭

💫 Author 💫

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

💖 subroutine chbev 💖

subroutine chbev (character JOBZ, character UPLO, integer N, integer KD,
    complex, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) W,
    complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * )
    WORK, real, dimension( * ) RWORK, integer INFO)

CHBEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices 🌸

🌟 Purpose 🌟

CHBEV computes all the eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix A. yes, all of them. she's thorough 💅

🎀 Parameters 🎀

JOBZ — JOBZ is CHARACTER*1.
- = 'N': compute eigenvalues only;
- = 'V': compute eigenvalues and eigenvectors.
UPLO — UPLO is CHARACTER*1.
- = 'U': upper triangle of A is stored;
- = 'L': lower triangle of A is stored.
N — N is INTEGER. the order of the matrix A. N >= 0.
KD — KD is INTEGER. the number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.
AB — AB is COMPLEX array, dimension (LDAB, N).
- on entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first KD+1 rows of the array. the j-th column of A is stored in the j-th column of the array AB as follows:
  - if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
  - if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
- on exit, AB is overwritten by values generated during the reduction to tridiagonal form. if UPLO = 'U', the first superdiagonal and the diagonal of the tridiagonal matrix T are returned in rows KD and KD+1 of AB, and if UPLO = 'L', the diagonal and first subdiagonal of T are returned in the first two rows of AB.
LDAB — LDAB is INTEGER. the leading dimension of the array AB. LDAB >= KD + 1.
W — W is REAL array, dimension (N). if INFO = 0, the eigenvalues in ascending order. ⭐
Z — Z is COMPLEX array, dimension (LDZ, N).
- if JOBZ = 'V', then if INFO = 0, Z contains the orthonormal eigenvectors of the matrix A, with the i-th column of Z holding the eigenvector associated with W(i).
- if JOBZ = 'N', then Z is not referenced. she's not even invited to the function call.
LDZ — LDZ is INTEGER. the leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
WORK — WORK is COMPLEX array, dimension (N).
RWORK — RWORK is REAL array, dimension (max(1,3*N-2)).
INFO — okay so the INFO tea continues in the next part bestie, don't leave me hanging 🫶💖
INFO is INTEGER — the vibe check at the end, bestie 💅
- = 0: successful exit. slay, we ate 🎀
- < 0: if INFO = -i, the i-th argument had an illegal value. someone handed us a bad arg, rude.
- > 0: if INFO = i, the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero. it tried, it really did 😭

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine `chbev_2stage`

subroutine chbev_2stage (character JOBZ, character UPLO, integer N, integer
    KD, complex, dimension( ldab, * ) AB, integer LDAB, real, dimension( * )
    W, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * )
    WORK, integer LWORK, real, dimension( * ) RWORK, integer INFO)

CHBEV_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices 🦋

Purpose

CHBEV_2STAGE computes all the eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix A using the 2stage technique for the reduction to tridiagonal. two-stage pipeline, main character energy ✨

Parameters

JOBZ — is CHARACTER*1
- = 'N': Compute eigenvalues only;
- = 'V': Compute eigenvalues and eigenvectors. Not available in this release. so close yet so far 🙈
UPLO — is CHARACTER*1
- = 'U': Upper triangle of A is stored;
- = 'L': Lower triangle of A is stored.
N — is INTEGER. The order of the matrix A. N >= 0.
KD — is INTEGER. The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.
AB — is COMPLEX array, dimension (LDAB, N).

On entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows:

if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).

On exit, AB is overwritten by values generated during the reduction to tridiagonal form. If UPLO = 'U', the first superdiagonal and the diagonal of the tridiagonal matrix T are returned in rows KD and KD+1 of AB, and if UPLO = 'L', the diagonal and first subdiagonal of T are returned in the first two rows of AB. it literally reinvents itself, growth 💖

LDAB — is INTEGER. The leading dimension of the array AB. LDAB >= KD + 1.
W — is REAL array, dimension (N). If INFO = 0, the eigenvalues in ascending order.
Z — is COMPLEX array, dimension (LDZ, N). If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal eigenvectors of the matrix A, with the i-th column of Z holding the eigenvector associated with W(i). If JOBZ = 'N', then Z is not referenced. ghosted, respectfully 👻
LDZ — is INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
WORK — is COMPLEX array, dimension LWORK. On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK — is INTEGER. The length of the array WORK. LWORK >= 1, when N <= 1; otherwise

If JOBZ = 'N' and N > 1, LWORK must be queried.

LWORK = MAX(1, dimension) where
dimension = (2KD+1)*N + KD*NTHREADS
where KD is the size of the band.
NTHREADS is the number of threads used when
openMP compilation is enabled, otherwise =1.

If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available.

If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.

RWORK — is REAL array, dimension (max(1,3*N-2))
INFO — is INTEGER
- = 0: successful exit.
- < 0: if INFO = -i, the i-th argument had an illegal value.
- > 0: if INFO = i, the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details

All details about the 2stage techniques are available in:

Azzam Haidar, Hatem Ltaief, and Jack Dongarra. Parallel reduction to condensed forms for symmetric eigenvalue problems using aggregated fine-grained and memory-aware kernels. In Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '11), New York, NY, USA, Article 8 , 11 pages. http://doi.acm.org/10.1145/2063384.2063394

A. Haidar, J. Kurzak, P. Luszczek, 2013. An improved parallel singular value algorithm and its implementation for multicore hardware, In Proceedings of 2013 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '13). Denver, Colorado, USA, 2013. Article 90, 12 pages. http://doi.acm.org/10.1145/2503210.2503292

A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra. A novel hybrid CPU-GPU generalized eigensolver for electronic structure calculations based on fine-grained memory aware tasks. International Journal of High Performance Computing Applications. Volume 28 Issue 2, Pages 196-209, May 2014. http://hpc.sagepub.com/content/28/2/196

subroutine `chbevd`

subroutine chbevd (character JOBZ, character UPLO, integer N, integer KD,
    complex, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) W,
    complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * )
    WORK, integer LWORK, real, dimension( * ) RWORK, integer LRWORK,
    integer, dimension( * ) IWORK, integer LIWORK, integer INFO)

CHBEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices 🌈

Purpose

CHBEVD computes all the eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix A. If eigenvectors are desired, it uses a divide and conquer algorithm. divide and conquer? she said divide and slay 👑

The divide and conquer algorithm makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none. so like, no cap, we've never actually seen it break 🫶

Parameters

JOBZ — is CHARACTER*1
- = 'N': Compute eigenvalues only;
- = 'V': Compute eigenvalues and eigenvectors.
UPLO — is CHARACTER*1
- = 'U': Upper triangle of A is stored;
- = 'L': Lower triangle of A is stored.
N — is INTEGER. The order of the matrix A. N >= 0.
KD — is INTEGER. The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.
AB — is COMPLEX array, dimension (LDAB, N).

On entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows:

if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).

LDAB — is INTEGER. The leading dimension of the array AB. LDAB >= KD + 1.
W — is REAL array, dimension (N). If INFO = 0, the eigenvalues in ascending order.
Z — is COMPLEX array, dimension (LDZ, N). If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal eigenvectors of the matrix A, with the i-th column of Z holding the eigenvector associated with W(i). If JOBZ = 'N', then Z is not referenced.
LDZ — is INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
WORK — is COMPLEX array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK — is INTEGER. The dimension of the array WORK.
- If N <= 1, LWORK must be at least 1.
- If JOBZ = 'N' and N > 1, LWORK must be at least N.
- If JOBZ = 'V' and N > 1, LWORK must be at least 2*N**2.

RWORK — is REAL array, dimension (LRWORK) On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.
LRWORK
LRWORK — INTEGER, the dimension of array RWORK. Bestie, this is how much room your real workspace gets, and the rules are:
- If N <= 1, LRWORK must be at least 1.
- If JOBZ = 'N' and N > 1, LRWORK must be at least N.
- If JOBZ = 'V' and N > 1, LRWORK must be at least 1 + 5*N + 2*N**2.

And if LRWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA. 🫶

IWORK — IWORK is INTEGER array, dimension (MAX(1,LIWORK)). On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. ✨
LIWORK — INTEGER, the dimension of array IWORK:
- If JOBZ = 'N' or N <= 1, LIWORK must be at least 1.
- If JOBZ = 'V' and N > 1, LIWORK must be at least 3 + 5*N .

If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA. Same deal, different integer. 😌

INFO — INTEGER, the vibe report:
- = 0: successful exit. 🎉
- < 0: if INFO = -i, the i-th argument had an illegal value.
- > 0: if INFO = i, the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero.

ngl, INFO > 0 is the algorithm ghosting you mid-computation. It literally did not converge. Not iconic, but at least it tells you exactly how many off-diagonal elements refused to behave.

🎀 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🦋 subroutine `chbevd_2stage`

subroutine chbevd_2stage (character JOBZ, character UPLO, integer N, integer
    KD, complex, dimension( ldab, * ) AB, integer LDAB, real, dimension( * )
    W, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * )
    WORK, integer LWORK, real, dimension( * ) RWORK, integer LRWORK,
    integer, dimension( * ) IWORK, integer LIWORK, integer INFO)

CHBEVD_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices 💅

✨ Purpose

CHBEVD_2STAGE computes all the eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix A using the 2stage technique for the reduction to tridiagonal. If eigenvectors are desired, it uses a divide and conquer algorithm. Two techniques, one glow-up. 👑

The divide and conquer algorithm makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none. Like, it could fail, but nobody's ever seen it happen. Fr fr.

🌸 Parameters

JOBZ — JOBZ is CHARACTER*1
- = 'N': Compute eigenvalues only;
- = 'V': Compute eigenvalues and eigenvectors. Not available in this release.

Heads up bestie: 'V' is literally not available in this release. Don't come for me when it doesn't do the thing.

UPLO — UPLO is CHARACTER*1
- = 'U': Upper triangle of A is stored;
- = 'L': Lower triangle of A is stored.
N — N is INTEGER. The order of the matrix A. N >= 0.
KD — KD is INTEGER. The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.
AB — AB is COMPLEX array, dimension (LDAB, N). On entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows:
- if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
- if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).

On exit, AB is overwritten by values generated during the reduction to tridiagonal form. If UPLO = 'U', the first superdiagonal and the diagonal of the tridiagonal matrix T are returned in rows KD and KD+1 of AB, and if UPLO = 'L', the diagonal and first subdiagonal of T are returned in the first two rows of AB. So your input array is gone-gone. It's giving "no takebacks." 🙈

LDAB — LDAB is INTEGER. The leading dimension of the array AB. LDAB >= KD + 1.
W — W is REAL array, dimension (N). If INFO = 0, the eigenvalues in ascending order. ⭐
Z — Z is COMPLEX array, dimension (LDZ, N). If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal eigenvectors of the matrix A, with the i-th column of Z holding the eigenvector associated with W(i). If JOBZ = 'N', then Z is not referenced. Not referenced means we don't even look at it. It's giving "you're not on the guest list."
LDZ — LDZ is INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
WORK — WORK is COMPLEX array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK — LWORK is INTEGER. The length of the array WORK. LWORK >= 1, when N <= 1; otherwise:
- If JOBZ = 'N' and N > 1, LWORK must be queried. LWORK = MAX(1, dimension) where dimension = (2KD+1)*N + KD*NTHREADS where KD is the size of the band. NTHREADS is the number of threads used when openMP compilation is enabled, otherwise =1.
- If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available.

Pro move: query first, allocate second. That's the whole game with LWORK = -1. 🧚

RWORK — RWORK is REAL array, dimension (LRWORK) On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.
LRWORK — LRWORK is INTEGER. The dimension of array RWORK.
- If N <= 1, LRWORK must be at least 1.
- If JOBZ = 'N' and N > 1, LRWORK must be at least N.
- If JOBZ = 'V' and N > 1, LRWORK must be at least 1 + 5*N + 2*N**2.

If LRWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.

IWORK — IWORK is INTEGER array, dimension (MAX(1,LIWORK)). On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
LIWORK — LIWORK is INTEGER. The dimension of array IWORK.
- If JOBZ = 'N' or N <= 1, LIWORK must be at least 1.
- If JOBZ = 'V' and N > 1, LIWORK must be at least 3 + 5*N .

INFO — INFO is INTEGER
- = 0: successful exit. ✅
- < 0: if INFO = -i, the i-th argument had an illegal value.
- > 0: if INFO = i, the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero.

💖 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🔥 Further Details

All details about the 2stage techniques are available in:

🫶 subroutine `chbevx`

subroutine chbevx (character JOBZ, character RANGE, character UPLO, integer
    N, integer KD, complex, dimension( ldab, * ) AB, integer LDAB, complex,
    dimension( ldq, * ) Q, integer LDQ, real VL, real VU, integer IL,
    integer IU, real ABSTOL, integer M, real, dimension( * ) W, complex,
    dimension( ldz, * ) Z, integer LDZ, complex, dimension( * ) WORK, real,
    dimension( * ) RWORK, integer, dimension( * ) IWORK, integer, dimension(
    * ) IFAIL, integer INFO)

CHBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices 🌈

✨ Purpose

CHBEVX computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix A. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. You get to choose. Main character energy, honestly. 🎀

🌸 Parameters

JOBZ
JOBZ is CHARACTER*1 ✨
- = 'N': compute eigenvalues only, that's it;
- = 'V': compute eigenvalues and eigenvectors.
RANGE is CHARACTER*1 🌈
- = 'A': every single eigenvalue is getting found, no filter;
- = 'V': only the eigenvalues chillin' in the half-open interval (VL,VU] get found;
- = 'I': the IL-th through IU-th eigenvalues get found.
UPLO is CHARACTER*1 💅
- = 'U': the upper triangle of A is what's stored;
- = 'L': the lower triangle of A is what's stored.
N is INTEGER
- the order of the matrix A. N >= 0.
KD is INTEGER
- the number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.
AB is COMPLEX array, dimension (LDAB, N) 🦋
- on entry, it's holding the upper or lower triangle of the Hermitian band matrix A, stored in the first KD+1 rows of the array. The j-th column of A lives in the j-th column of the array AB, like this:

if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

on exit, AB gets overwritten by values generated during the reduction to tridiagonal form. she's been through something, bestie 😭

LDAB is INTEGER
- the leading dimension of the array AB. LDAB >= KD + 1.
Q is COMPLEX array, dimension (LDQ, N) 🎀
- if JOBZ = 'V', this is the N-by-N unitary matrix used in the reduction to tridiagonal form.
- if JOBZ = 'N', the array Q is not referenced at all, she's off the clock.
LDQ is INTEGER
- the leading dimension of the array Q. If JOBZ = 'V', then LDQ >= max(1,N).
VL is REAL
- if RANGE='V', this is the lower bound of the interval to be searched for eigenvalues. VL < VU.
- not referenced if RANGE = 'A' or 'I'.
VU is REAL
- if RANGE='V', this is the upper bound of the interval to be searched for eigenvalues. VL < VU.
- not referenced if RANGE = 'A' or 'I'.
IL is INTEGER
- if RANGE='I', this is the index of the smallest eigenvalue to be returned.
- 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
- not referenced if RANGE = 'A' or 'V'.
IU is INTEGER
- if RANGE='I', this is the index of the largest eigenvalue to be returned.
- 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
- not referenced if RANGE = 'A' or 'V'.
ABSTOL is REAL ⭐
- the absolute error tolerance for the eigenvalues. An approximate eigenvalue gets accepted as converged when it's determined to lie in an interval [a,b] of width less than or equal to

        ABSTOL + EPS *   max( |a|,|b| ) ,

where EPS is the machine precision. If ABSTOL is less than or equal to zero, then EPS*|T| will be used in its place, where |T| is the 1-norm of the tridiagonal matrix obtained by reducing AB to tridiagonal form.
eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*SLAMCH('S'), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*SLAMCH('S'). like, that's the fix, fr fr.
see 'Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High Relative Accuracy,' by Demmel and Kahan, LAPACK Working Note #3.

M is INTEGER
- the total number of eigenvalues found. 0 <= M <= N.
- if RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
W is REAL array, dimension (N) 💖
- the first M elements contain the selected eigenvalues in ascending order. organized queen.
Z is COMPLEX array, dimension (LDZ, max(1,M)) 👑
- if JOBZ = 'V', then if INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i).
- if an eigenvector fails to converge, then that column of Z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in IFAIL.
- if JOBZ = 'N', then Z is not referenced.
- note: the user must ensure that at least max(1,M) columns are supplied in the array Z; if RANGE = 'V', the exact value of M is not known in advance and an upper bound must be used. so plan ahead, bestie 🙈
LDZ is INTEGER
- the leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
WORK is COMPLEX array, dimension (N)
RWORK is REAL array, dimension (7*N)
IWORK is INTEGER array, dimension (5*N)
IFAIL is INTEGER array, dimension (N) 💀
- if JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL are zero. If INFO > 0, then IFAIL contains the indices of the eigenvectors that failed to converge.
- if JOBZ = 'N', then IFAIL is not referenced.
INFO is INTEGER
- = 0: successful exit, we ate;
- < 0: if INFO = -i, the i-th argument had an illegal value;
- > 0: if INFO = i, then i eigenvectors failed to converge. Their indices are stored in array IFAIL.

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🧚 `subroutine chbevx_2stage`

subroutine chbevx_2stage (character JOBZ, character RANGE, character UPLO,
    integer N, integer KD, complex, dimension( ldab, * ) AB, integer LDAB,
    complex, dimension( ldq, * ) Q, integer LDQ, real VL, real VU, integer
    IL, integer IU, real ABSTOL, integer M, real, dimension( * ) W, complex,
    dimension( ldz, * ) Z, integer LDZ, complex, dimension( * ) WORK,
    integer LWORK, real, dimension( * ) RWORK, integer, dimension( * )
    IWORK, integer, dimension( * ) IFAIL, integer INFO)

CHBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices ✨

🎯 Purpose:

CHBEVX_2STAGE computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix A using the 2stage technique for the reduction to tridiagonal. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. pick your filter, she's flexible like that 💅

🌸 Parameters

JOBZ is CHARACTER*1
- = 'N': compute eigenvalues only;
- = 'V': compute eigenvalues and eigenvectors. Not available in this release. 😭 so close yet so far
RANGE is CHARACTER*1 🌈
- = 'A': all eigenvalues will be found;
- = 'V': all eigenvalues in the half-open interval (VL,VU] will be found;
- = 'I': the IL-th through IU-th eigenvalues will be found.
UPLO is CHARACTER*1 💅
- = 'U': upper triangle of A is stored;
- = 'L': lower triangle of A is stored.
N is INTEGER
- the order of the matrix A. N >= 0.
KD is INTEGER
- the number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.
AB is COMPLEX array, dimension (LDAB, N) 🦋
- on entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows:

if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

on exit, AB is overwritten by values generated during the reduction to tridiagonal form.

LDAB is INTEGER
- the leading dimension of the array AB. LDAB >= KD + 1.
Q is COMPLEX array, dimension (LDQ, N) 🎀
- if JOBZ = 'V', the N-by-N unitary matrix used in the reduction to tridiagonal form.
- if JOBZ = 'N', the array Q is not referenced.
LDQ is INTEGER
- the leading dimension of the array Q. If JOBZ = 'V', then LDQ >= max(1,N).
VL is REAL
- if RANGE='V', the lower bound of the interval to be searched for eigenvalues. VL < VU.
- not referenced if RANGE = 'A' or 'I'.
VU is REAL
- if RANGE='V', the upper bound of the interval to be searched for eigenvalues. VL < VU.
- not referenced if RANGE = 'A' or 'I'.
IL is INTEGER
- if RANGE='I', the index of the smallest eigenvalue to be returned.
- 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
- not referenced if RANGE = 'A' or 'V'.
IU is INTEGER
- if RANGE='I', the index of the largest eigenvalue to be returned.
- 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
- not referenced if RANGE = 'A' or 'V'.
ABSTOL is REAL ⭐
- the absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to

        ABSTOL + EPS *   max( |a|,|b| ) ,

where EPS is the machine precision. If ABSTOL is less than or equal to zero, then EPS*|T| gets used in its place instead, where |T| is the 1-norm of the tridiagonal matrix you get by reducing AB to tridiagonal form. ✨

Eigenvalues get computed most accurately when ABSTOL is set to twice the underflow threshold 2*SLAMCH('S'), not zero. So if this routine comes back with INFO>0, which means some eigenvectors did not converge, bestie, try setting ABSTOL to 2*SLAMCH('S'). 🫶

See 'Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High Relative Accuracy,' by Demmel and Kahan, LAPACK Working Note #3. iykyk 📚

M — M is INTEGER. The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', then M = N, and if RANGE = 'I', then M = IU-IL+1. 💅
W — W is REAL array, dimension (N). The first M elements hold the selected eigenvalues in ascending order. So cute, so tidy.
Z — Z is COMPLEX array, dimension (LDZ, max(1,M)). If JOBZ = 'V', then if INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i). If an eigenvector fails to converge, then that column of Z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in IFAIL. If JOBZ = 'N', then Z is not referenced. 🚫

Note: the user must ensure that at least max(1,M) columns are supplied in the array Z; if RANGE = 'V', the exact value of M is not known in advance and an upper bound must be used. 👀

LDZ — LDZ is INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
WORK — WORK is COMPLEX array, dimension (LWORK).
LWORK — LWORK is INTEGER. The length of the array WORK. LWORK >= 1, when N <= 1; otherwise:

If JOBZ = 'N' and N > 1, LWORK must be queried.
                         LWORK = MAX(1, dimension) where
                         dimension = (2KD+1)*N + KD*NTHREADS
                         where KD is the size of the band.
                         NTHREADS is the number of threads used when
                         openMP compilation is enabled, otherwise =1.
If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available.

RWORK — RWORK is REAL array, dimension (7*N).
IWORK — IWORK is INTEGER array, dimension (5*N).
IFAIL — IFAIL is INTEGER array, dimension (N). If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL are zero. If INFO > 0, then IFAIL contains the indices of the eigenvectors that failed to converge. If JOBZ = 'N', then IFAIL is not referenced. 🎀
INFO — INFO is INTEGER.
- = 0: successful exit ✨
- < 0: if INFO = -i, the i-th argument had an illegal value 😭
- > 0: if INFO = i, then i eigenvectors failed to converge. Their indices are stored in array IFAIL. 💀

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🔍 Further Details:

All details about the 2stage techniques are available in:

🦋 CHBGV

subroutine chbgv (character JOBZ, character UPLO, integer N, integer KA,
    integer KB, complex, dimension( ldab, * ) AB, integer LDAB, complex,
    dimension( ldbb, * ) BB, integer LDBB, real, dimension( * ) W, complex,
    dimension( ldz, * ) Z, integer LDZ, complex, dimension( * ) WORK, real,
    dimension( * ) RWORK, integer INFO)

🎯 Purpose:

CHBGV computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite banded eigenproblem, of the form A*x=(lambda)*B*x. Here A and B are assumed to be Hermitian and banded, and B is also positive definite. periodt 💅

🌸 Parameters

JOBZ — JOBZ is CHARACTER*1.
- = 'N': Compute eigenvalues only;
- = 'V': Compute eigenvalues and eigenvectors.
UPLO — UPLO is CHARACTER*1.
- = 'U': Upper triangles of A and B are stored;
- = 'L': Lower triangles of A and B are stored.
N — N is INTEGER. The order of the matrices A and B. N >= 0.
KA — KA is INTEGER. The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KA >= 0.
KB — KB is INTEGER. The number of superdiagonals of the matrix B if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KB >= 0.
AB — AB is COMPLEX array, dimension (LDAB, N). On entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first ka+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows:

if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).

On exit, the contents of AB are destroyed. gone. reduced to atoms. 💀

LDAB — LDAB is INTEGER. The leading dimension of the array AB. LDAB >= KA+1.
BB — BB is COMPLEX array, dimension (LDBB, N). On entry, the upper or lower triangle of the Hermitian band matrix B, stored in the first kb+1 rows of the array. The j-th column of B is stored in the j-th column of the array BB as follows:

if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).

On exit, the factor S from the split Cholesky factorization B = S**H*S, as returned by CPBSTF.

LDBB — LDBB is INTEGER. The leading dimension of the array BB. LDBB >= KB+1.
W — W is REAL array, dimension (N). If INFO = 0, the eigenvalues in ascending order. 🎀
Z — Z is COMPLEX array, dimension (LDZ, N). If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of eigenvectors, with the i-th column of Z holding the eigenvector associated with W(i). The eigenvectors are normalized so that Z**H*B*Z = I. If JOBZ = 'N', then Z is not referenced.
LDZ — LDZ is INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= N.
WORK — WORK is COMPLEX array, dimension (N).
RWORK — RWORK is REAL array, dimension (3*N).
INFO — INFO is INTEGER.
- = 0: successful exit ✨
- < 0: if INFO = -i, the i-th argument had an illegal value
- > 0: if INFO = i, and i is:
  - <= N: the algorithm failed to converge: i off-diagonal elements of an intermediate tridiagonal form did not converge to zero;
  - > N: if INFO = N + i, for 1 <= i <= N, then CPBSTF returned INFO = i: B is not positive definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed. 💀

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🌈 CHBGVD

subroutine chbgvd (character JOBZ, character UPLO, integer N, integer KA,
    integer KB, complex, dimension( ldab, * ) AB, integer LDAB, complex,
    dimension( ldbb, * ) BB, integer LDBB, real, dimension( * ) W, complex,
    dimension( ldz, * ) Z, integer LDZ, complex, dimension( * ) WORK,
    integer LWORK, real, dimension( * ) RWORK, integer LRWORK, integer,
    dimension( * ) IWORK, integer LIWORK, integer INFO)

🎯 Purpose:

CHBGVD computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite banded eigenproblem, of the form A*x=(lambda)*B*x. Here A and B are assumed to be Hermitian and banded, and B is also positive definite. If eigenvectors are desired, it uses a divide and conquer algorithm. main character energy fr fr 👑

okay so the divide and conquer algorithm is literally the most chill about floating point arithmetic — it makes very mild assumptions, like, barely any. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but bestie, we know of none. 💅

✨ Parameters ✨

JOBZ — this one is a CHARACTER*1, okay?
- = 'N': just compute eigenvalues only, nothing else, periodt.
- = 'V': compute eigenvalues and eigenvectors, the full slay.
UPLO — also CHARACTER*1, telling us which triangles are stored.
- = 'U': upper triangles of A and B are stored.
- = 'L': lower triangles of A and B are stored.
N — INTEGER, the order of the matrices A and B. N >= 0.
KA — INTEGER, the number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KA >= 0.
KB — INTEGER, the number of superdiagonals of the matrix B if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KB >= 0.
AB — COMPLEX array, dimension (LDAB, N). On entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first ka+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows:
- if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
- if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka). On exit, the contents of AB are destroyed. (rip AB, you served 💀)
LDAB — INTEGER, the leading dimension of the array AB. LDAB >= KA+1.
BB — COMPLEX array, dimension (LDBB, N). On entry, the upper or lower triangle of the Hermitian band matrix B, stored in the first kb+1 rows of the array. The j-th column of B is stored in the j-th column of the array BB as follows:
- if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
- if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb). On exit, the factor S from the split Cholesky factorization B = S**H*S, as returned by CPBSTF.
LDBB — INTEGER, the leading dimension of the array BB. LDBB >= KB+1.
W — REAL array, dimension (N). If INFO = 0, the eigenvalues in ascending order.
Z — COMPLEX array, dimension (LDZ, N). If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of eigenvectors, with the i-th column of Z holding the eigenvector associated with W(i). The eigenvectors are normalized so that Z**H*B*Z = I. If JOBZ = 'N', then Z is not referenced.
LDZ — INTEGER, the leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= N.
WORK — COMPLEX array, dimension (MAX(1,LWORK)). On exit, if INFO=0, WORK(1) returns the optimal LWORK.
LWORK — INTEGER, the dimension of the array WORK.
- If N <= 1, LWORK >= 1.
- If JOBZ = 'N' and N > 1, LWORK >= N.
- If JOBZ = 'V' and N > 1, LWORK >= 2*N**2.

RWORK — REAL array, dimension (MAX(1,LRWORK)). On exit, if INFO=0, RWORK(1) returns the optimal LRWORK.
LRWORK — INTEGER, the dimension of array RWORK.
- If N <= 1, LRWORK >= 1.
- If JOBZ = 'N' and N > 1, LRWORK >= N.
- If JOBZ = 'V' and N > 1, LRWORK >= 1 + 5*N + 2*N**2.

IWORK — INTEGER array, dimension (MAX(1,LIWORK)). On exit, if INFO=0, IWORK(1) returns the optimal LIWORK.
LIWORK — INTEGER, the dimension of array IWORK.
- If JOBZ = 'N' or N <= 1, LIWORK >= 1.
- If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N.

INFO — INTEGER.
- = 0: successful exit.
- < 0: if INFO = -i, the i-th argument had an illegal value.
- > 0: if INFO = i, and i is:
  - <= N: the algorithm failed to converge: i off-diagonal elements of an intermediate tridiagonal form did not converge to zero;
  - > N: if INFO = N + i, for 1 <= i <= N, then CPBSTF returned INFO = i: B is not positive definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed.

👑 Author 👑

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

🫶 Contributors: 🫶

Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

🌸 subroutine chbgvx 🌸

subroutine chbgvx (character JOBZ, character RANGE, character UPLO, integer
    N, integer KA, integer KB, complex, dimension( ldab, * ) AB, integer
    LDAB, complex, dimension( ldbb, * ) BB, integer LDBB, complex,
    dimension( ldq, * ) Q, integer LDQ, real VL, real VU, integer IL,
    integer IU, real ABSTOL, integer M, real, dimension( * ) W, complex,
    dimension( ldz, * ) Z, integer LDZ, complex, dimension( * ) WORK, real,
    dimension( * ) RWORK, integer, dimension( * ) IWORK, integer, dimension(
    * ) IFAIL, integer INFO)
CHBGVX

🎯 Purpose 🎯

CHBGVX computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite banded eigenproblem, of the form A*x=(lambda)*B*x. Here A and B are assumed to be Hermitian and banded, and B is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either all eigenvalues, a range of values or a range of indices for the desired eigenvalues.

✨ Parameters ✨

JOBZ — CHARACTER*1
- = 'N': compute eigenvalues only;
- = 'V': compute eigenvalues and eigenvectors.
RANGE — CHARACTER*1
- = 'A': all eigenvalues will be found;
- = 'V': all eigenvalues in the half-open interval (VL,VU] will be found;
- = 'I': the IL-th through IU-th eigenvalues will be found.
UPLO — CHARACTER*1
- = 'U': upper triangles of A and B are stored;
- = 'L': lower triangles of A and B are stored.
N — INTEGER, the order of the matrices A and B. N >= 0.
KA — INTEGER, the number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KA >= 0.
KB — INTEGER, the number of superdiagonals of the matrix B if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KB >= 0.
AB — COMPLEX array, dimension (LDAB, N). On entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first ka+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows:
- if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
- if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka). On exit, the contents of AB are destroyed.
LDAB — INTEGER, the leading dimension of the array AB. LDAB >= KA+1.
BB — COMPLEX array, dimension (LDBB, N). On entry, the upper or lower triangle of the Hermitian band matrix B, stored in the first kb+1 rows of the array. The j-th column of B is stored in the j-th column of the array BB as follows:
- if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
- if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb). On exit, the factor S from the split Cholesky factorization B = S**H*S, as returned by CPBSTF.
LDBB — INTEGER, the leading dimension of the array BB. LDBB >= KB+1.
Q — COMPLEX array, dimension (LDQ, N). If JOBZ = 'V', the n-by-n matrix used in the reduction of A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x, and consequently C to tridiagonal form. If JOBZ = 'N', the array Q is not referenced.
LDQ — INTEGER, the leading dimension of the array Q. If JOBZ = 'N', LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N).
VL — REAL. If RANGE='V', the lower bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.
VU — REAL. If RANGE='V', the upper bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.
IL —

`IL` 💜

IL is INTEGER

If RANGE='I', the index of the smallest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.

`IU` ✨

IU is INTEGER

If RANGE='I', the index of the largest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.

`ABSTOL` 🎀

ABSTOL is REAL

The absolute error tolerance for the eigenvalues, bestie. An approximate eigenvalue gets accepted as converged when it's determined to lie in an interval [a,b] of width less than or equal to

ABSTOL + EPS *   max( |a|,|b| ) ,

where EPS is the machine precision. If ABSTOL is less than or equal to zero, then EPS*|T| will be used in its place, where |T| is the 1-norm of the tridiagonal matrix obtained by reducing AP to tridiagonal form.

Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*SLAMCH('S'), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*SLAMCH('S').

fr fr, don't set it to zero — set it to 2*SLAMCH('S') if you want the most accurate glow-up 💅

`M` 🌸

M is INTEGER

The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

`W` 🦋

W is REAL array, dimension (N)

If INFO = 0, the eigenvalues in ascending order. Sorted and everything, slay.

`Z` 💅

Z is COMPLEX array, dimension (LDZ, N)

If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of eigenvectors, with the i-th column of Z holding the eigenvector associated with W(i). The eigenvectors are normalized so that Z**H*B*Z = I. If JOBZ = 'N', then Z is not referenced.

`LDZ` 🔥

LDZ is INTEGER

The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= N.

`WORK`

WORK is COMPLEX array, dimension (N)

`RWORK`

RWORK is REAL array, dimension (7*N)

`IWORK`

IWORK is INTEGER array, dimension (5*N)

`IFAIL` 👑

IFAIL is INTEGER array, dimension (N)

If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL are zero. If INFO > 0, then IFAIL contains the indices of the eigenvectors that failed to converge. If JOBZ = 'N', then IFAIL is not referenced.

`INFO` 💀

INFO is INTEGER

= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value
> 0:  if INFO = i, and i is:
   <= N:  then i eigenvectors failed to converge.  Their
          indices are stored in array IFAIL.
   > N:   if INFO = N + i, for 1 <= i <= N, then CPBSTF
          returned INFO = i: B is not positive definite.
          The factorization of B could not be completed and
          no eigenvalues or eigenvectors were computed.

the > N case is the sad one — B isn't positive definite, the factorization of B couldn't be finished, and no eigenvalues or eigenvectors were computed at all 😭 nothing came through, zero, nada.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

`subroutine chpev` ✨

subroutine chpev (character JOBZ, character UPLO, integer N, complex,
    dimension( * ) AP, real, dimension( * ) W, complex, dimension( ldz, * )
    Z, integer LDZ, complex, dimension( * ) WORK, real, dimension( * )
    RWORK, integer INFO)

CHPEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

CHPEV computes all the eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix in packed storage.

Parameters

JOBZ 🎀

JOBZ is CHARACTER*1

= 'N':  Compute eigenvalues only;
= 'V':  Compute eigenvalues and eigenvectors.

UPLO 🌸

UPLO is CHARACTER*1

= 'U':  Upper triangle of A is stored;
= 'L':  Lower triangle of A is stored.

N 🦋

N is INTEGER

The order of the matrix A. N >= 0.

AP 💜

AP is COMPLEX array, dimension (N*(N+1)/2)

On entry, the upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows:

if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

On exit, AP is overwritten by values generated during the reduction to tridiagonal form. If UPLO = 'U', the diagonal and first superdiagonal of the tridiagonal matrix T overwrite the corresponding elements of A, and if UPLO = 'L', the diagonal and first subdiagonal of T overwrite the corresponding elements of A.

W ✨

W is REAL array, dimension (N)

If INFO = 0, the eigenvalues in ascending order.

Z 💅

Z is COMPLEX array, dimension (LDZ, N)

If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal eigenvectors of the matrix A, with the i-th column of Z holding the eigenvector associated with W(i). If JOBZ = 'N', then Z is not referenced.

LDZ 🔥

LDZ is INTEGER

The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).

WORK

WORK is COMPLEX array, dimension (max(1, 2*N-1))

RWORK

RWORK is REAL array, dimension (max(1, 3*N-2))

INFO 💀

INFO is INTEGER

= 0:  successful exit.
< 0:  if INFO = -i, the i-th argument had an illegal value.
> 0:  if INFO = i, the algorithm failed to converge; i
      off-diagonal elements of an intermediate tridiagonal
      form did not converge to zero.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

`subroutine chpevd` 🌈

subroutine chpevd (character JOBZ, character UPLO, integer N, complex,
    dimension( * ) AP, real, dimension( * ) W, complex, dimension( ldz, * )
    Z, integer LDZ, complex, dimension( * ) WORK, integer LWORK, real,
    dimension( * ) RWORK, integer LRWORK, integer, dimension( * ) IWORK,
    integer LIWORK, integer INFO)

CHPEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

CHPEVD computes all the eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A in packed storage. If eigenvectors are desired, it uses a divide and conquer algorithm.

no cap, the divide and conquer vibe is mild about floating point. Cray girlies, you're fine 💖 hex/decimal machines without guard digits, maybe not — but we've literally never met one.

Parameters

JOBZ 🎀

JOBZ is CHARACTER*1

= 'N':  Compute eigenvalues only;
= 'V':  Compute eigenvalues and eigenvectors.

UPLO 🌸

UPLO is CHARACTER*1

= 'U':  Upper triangle of A is stored;
= 'L':  Lower triangle of A is stored.

N 🦋

N is INTEGER

The order of the matrix A. N >= 0.

AP 💜

AP is COMPLEX array, dimension (N*(N+1)/2)

On entry, the upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows:

if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

W ✨

W is REAL array, dimension (N)

If INFO = 0, the eigenvalues in ascending order.

Z 💅

Z is COMPLEX array, dimension (LDZ, N)

LDZ 🔥

LDZ is INTEGER

The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).

WORK ⭐

WORK is COMPLEX array, dimension (MAX(1,LWORK))

On exit, if INFO = 0, WORK(1) returns the required LWORK.

LWORK 👑

LWORK is INTEGER

The dimension of array WORK.

If N <= 1,               LWORK must be at least 1.
If JOBZ = 'N' and N > 1, LWORK must be at least N.
If JOBZ = 'V' and N > 1, LWORK must be at least 2*N.

If LWORK = -1, then a workspace query is assumed; the routine only calculates the required sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.

LWORK = -1 is the "bestie, just tell me how big" mode — it peeks at the sizes, hands them back, and XERBLA stays completely quiet about it 🤫

RWORK 🧚

RWORK is REAL array, dimension (MAX(1,LRWORK))

On exit, if INFO = 0, RWORK(1) returns the required LRWORK.

LRWORK

LRWORK — INTEGER, bestie, and it's the dimension of array RWORK. Get it wrong and it's over, so:
- If N <= 1, LRWORK must be at least 1.
- If JOBZ = 'N' and N > 1, LRWORK must be at least N.
- If JOBZ = 'V' and N > 1, LRWORK must be at least 1 + 5*N + 2*N**2.

And now the lil party trick: if LRWORK = -1, then a workspace query is assumed; the routine only calculates the required sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA. 💅 (No yelling from the error handler, we love to see it.)

IWORK — INTEGER array, dimension (MAX(1,LIWORK)). On exit, if INFO = 0, IWORK(1) returns the required LIWORK. It literally tells you what it needed, icon behaviour fr fr.
LIWORK — INTEGER. The dimension of array IWORK.
- If JOBZ = 'N' or N <= 1, LIWORK must be at least 1.
- If JOBZ = 'V' and N > 1, LIWORK must be at least 3 + 5*N.

If LIWORK = -1, then a workspace query is assumed; the routine only calculates the required sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA. Same deal as above, the -1 girlies all behave the same way.

INFO — INTEGER, the vibe report at the end:
- = 0: successful exit 🎉
- < 0: if INFO = -i, the i-th argument had an illegal value. Rude of you, honestly.
- > 0: if INFO = i, the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero. She tried her best and she could not.

🎀 Author 🎀

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🦋 subroutine `chpevx` 🦋

subroutine chpevx (character JOBZ, character RANGE, character UPLO, integer
    N, complex, dimension( * ) AP, real VL, real VU, integer IL, integer IU,
    real ABSTOL, integer M, real, dimension( * ) W, complex, dimension( ldz,
    * ) Z, integer LDZ, complex, dimension( * ) WORK, real, dimension( * )
    RWORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL,
    integer INFO)

CHPEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

✨ Purpose ✨

CHPEVX computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A in packed storage. Eigenvalues/vectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. Pick your fighters, basically — values OR indices, your call.

💖 Parameters 💖

JOBZ — CHARACTER*1
- = 'N': Compute eigenvalues only;
- = 'V': Compute eigenvalues and eigenvectors. (the full meal deal)
RANGE — CHARACTER*1
- = 'A': all eigenvalues will be found;
- = 'V': all eigenvalues in the half-open interval (VL,VU] will be found;
- = 'I': the IL-th through IU-th eigenvalues will be found.
UPLO — CHARACTER*1
- = 'U': Upper triangle of A is stored;
- = 'L': Lower triangle of A is stored.
N — INTEGER. The order of the matrix A. N >= 0.
AP — COMPLEX array, dimension (N*(N+1)/2). On entry, the upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows:

if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

VL — REAL. If RANGE='V', the lower bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.
VU — REAL. If RANGE='V', the upper bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.
IL — INTEGER. If RANGE='I', the index of the smallest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.
IU — INTEGER. If RANGE='I', the index of the largest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.
ABSTOL — REAL. The absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to

ABSTOL + EPS *   max( |a|,|b| ) ,

Big tip incoming: Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*SLAMCH('S'), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*SLAMCH('S'). Like, she's asking nicely, just do it.

See 'Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High Relative Accuracy,' by Demmel and Kahan, LAPACK Working Note #3.

M — INTEGER. The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
W — REAL array, dimension (N). If INFO = 0, the selected eigenvalues in ascending order. SORTED, she's organised, we stan.
Z — COMPLEX array, dimension (LDZ, max(1,M)). If JOBZ = 'V', then if INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i). If an eigenvector fails to converge, then that column of Z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in IFAIL. If JOBZ = 'N', then Z is not referenced.

Note: the user must ensure that at least max(1,M) columns are supplied in the array Z; if RANGE = 'V', the exact value of M is not known in advance and an upper bound must be used. So don't come crying when you under-allocated, bestie, the note warned you. 😭

LDZ — INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
WORK — COMPLEX array, dimension (2*N).
RWORK — REAL array, dimension (7*N).
IWORK — INTEGER array, dimension (5*N).
IFAIL — INTEGER array, dimension (N). If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL are zero. If INFO > 0, then IFAIL contains the indices of the eigenvectors that failed to converge. If JOBZ = 'N', then IFAIL is not referenced.
INFO — INTEGER:
- = 0: successful exit ✨
- < 0: if INFO = -i, the i-th argument had an illegal value
- > 0: if INFO = i, then i eigenvectors failed to converge. Their indices are stored in array IFAIL. She's telling you exactly which ones ghosted, at least.

🌸 Author 🌸

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🌈 subroutine `chpgv` 🌈

subroutine chpgv (integer ITYPE, character JOBZ, character UPLO, integer N,
    complex, dimension( * ) AP, complex, dimension( * ) BP, real, dimension(
    * ) W, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension(
    * ) WORK, real, dimension( * ) RWORK, integer INFO)

CHPGV

✨ Purpose ✨

CHPGV computes all the eigenvalues and, optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and B are assumed to be Hermitian, stored in packed format, and B is also positive definite. Two matrices, one problem, three possible shapes of it. Pick your form.

💅 Parameters 💅

ITYPE — INTEGER. Specifies the problem type to be solved:
- = 1: A*x = (lambda)*B*x
- = 2: A*B*x = (lambda)*x
- = 3: B*A*x = (lambda)*x
JOBZ — CHARACTER*1
- = 'N': Compute eigenvalues only;
- = 'V': Compute eigenvalues and eigenvectors.
UPLO — CHARACTER*1
- = 'U': Upper triangles of A and B are stored;
- = 'L': Lower triangles of A and B are stored.
N — INTEGER. The order of the matrices A and B. N >= 0.
AP — COMPLEX array, dimension (N*(N+1)/2). On entry, the upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows:

if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

On exit, the contents of AP are destroyed. GONE. reduced to atoms. do not expect it back. 💀

BP — COMPLEX array, dimension (N*(N+1)/2). On entry, the upper or lower triangle of the Hermitian matrix B, packed columnwise in a linear array. The j-th column of B is stored in the array BP as follows:

if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.

`W`

W is REAL array, dimension (N)

If INFO = 0, this is where the eigenvalues live, in ascending order ✨ like a little glow-up lineup, smallest to biggest, no notes.

`Z`

Z is COMPLEX array, dimension (LDZ, N)

If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of eigenvectors 🎀 and bestie, they come normalized in very specific ways:

if ITYPE = 1 or 2, Z**H*B*Z = I;
if ITYPE = 3, Z**H*inv(B)*Z = I.

If JOBZ = 'N', then Z is not referenced. It's giving "you're literally not invited to this computation" 💅

`LDZ`

LDZ is INTEGER

The leading dimension of the array Z. Rules are LDZ >= 1, and if JOBZ = 'V', then LDZ >= max(1,N).

`WORK`

WORK is COMPLEX array, dimension (max(1, 2*N-1))

`RWORK`

RWORK is REAL array, dimension (max(1, 3*N-2))

`INFO`

INFO is INTEGER

= 0: successful exit 🌈
< 0: if INFO = -i, the i-th argument had an illegal value 💀
> 0: CPPTRF or CHPEV returned an error code:
- <= N: if INFO = i, CHPEV failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not convergeto zero;
- > N: if INFO = N + i, for 1 <= i <= n, then the leading minor of order i of B is not positive definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed. so like, absolutely nothing was computed. zero. nada. the math simply said no 😭

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine `chpgvd`

subroutine chpgvd (integer ITYPE, character JOBZ, character UPLO, integer N,
    complex, dimension( * ) AP, complex, dimension( * ) BP, real, dimension(
    * ) W, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension(
    * ) WORK, integer LWORK, real, dimension( * ) RWORK, integer LRWORK,
    integer, dimension( * ) IWORK, integer LIWORK, integer INFO)

`CHPGVD` 💖

Purpose:

CHPGVD computes all the eigenvalues and, optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and B are assumed to be Hermitian, stored in packed format, and B is also positive definite. If eigenvectors are desired, it uses a divide and conquer algorithm 👑 (yes queen, divide and conquer, literally her whole brand).

The divide and conquer algorithm makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none. fr fr, "conceivably" is doing a LOT of heavy lifting in that sentence 🙈

Parameters

`ITYPE`

ITYPE is INTEGER

Specifies the problem type to be solved:

= 1: A*x = (lambda)*B*x
= 2: A*B*x = (lambda)*x
= 3: B*A*x = (lambda)*x

`JOBZ`

JOBZ is CHARACTER*1

= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.

`UPLO`

UPLO is CHARACTER*1

= 'U': Upper triangles of A and B are stored;
= 'L': Lower triangles of A and B are stored.

`N`

N is INTEGER

The order of the matrices A and B. N >= 0.

`AP`

AP is COMPLEX array, dimension (N*(N+1)/2)

On entry, the upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows:

if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

On exit, the contents of AP are destroyed. RIP AP, she was brave 🫡

`BP`

BP is COMPLEX array, dimension (N*(N+1)/2)

On entry, the upper or lower triangle of the Hermitian matrix B, packed columnwise in a linear array. The j-th column of B is stored in the array BP as follows:

if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.

On exit, the triangular factor U or L from the Cholesky factorization B = U**H*U or B = L*L**H, in the same storage format as B.

`W`

W is REAL array, dimension (N)

If INFO = 0, the eigenvalues in ascending order. sorted, bestie, you don't gotta do a thing ✨

`Z`

Z is COMPLEX array, dimension (LDZ, N)

If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of eigenvectors. The eigenvectors are normalized as follows:

if ITYPE = 1 or 2, Z**H*B*Z = I;
if ITYPE = 3, Z**H*inv(B)*Z = I.

If JOBZ = 'N', then Z is not referenced.

`LDZ`

LDZ is INTEGER

The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).

`WORK`

WORK is COMPLEX array, dimension (MAX(1,LWORK))

On exit, if INFO = 0, WORK(1) returns the required LWORK. it's like a little receipt for how much space you needed 🧾

`LWORK`

LWORK is INTEGER

The dimension of array WORK.

If N <= 1, LWORK >= 1.
If JOBZ = 'N' and N > 1, LWORK >= N.
If JOBZ = 'V' and N > 1, LWORK >= 2*N.

`RWORK`

RWORK is REAL array, dimension (MAX(1,LRWORK))

On exit, if INFO = 0, RWORK(1) returns the required LRWORK.

`LRWORK`

LRWORK is INTEGER

The dimension of array RWORK.

If N <= 1, LRWORK >= 1.
If JOBZ = 'N' and N > 1, LRWORK >= N.
If JOBZ = 'V' and N > 1, LRWORK >= 1 + 5*N + 2*N**2.

If LRWORK = -1, then a workspace query is assumed; the routine only calculates the required sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.

`IWORK`

IWORK is INTEGER array, dimension (MAX(1,LIWORK))

On exit, if INFO = 0, IWORK(1) returns the required LIWORK.

`LIWORK`

LIWORK is INTEGER

The dimension of array IWORK.

If JOBZ = 'N' or N <= 1, LIWORK >= 1.
If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N.

`INFO`

INFO is INTEGER

= 0: successful exit 🌈
< 0: if INFO = -i, the i-th argument had an illegal value 💀
> 0: CPPTRF or CHPEVD returned an error code:
- <= N: if INFO = i, CHPEVD failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not convergeto zero;
- > N: if INFO = N + i, for 1 <= i <= n, then the leading minor of order i of B is not positive definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed.

Heads up bestie: if B isn't actually positive definite, everything just stops. No eigenvalues, no eigenvectors, nothing. The math is not being flexible about this one, periodt.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA 🌸

subroutine `chpgvx`

subroutine chpgvx (integer ITYPE, character JOBZ, character RANGE, character
    UPLO, integer N, complex, dimension( * ) AP, complex, dimension( * ) BP,
    real VL, real VU, integer IL, integer IU, real ABSTOL, integer M, real,
    dimension( * ) W, complex, dimension( ldz, * ) Z, integer LDZ, complex,
    dimension( * ) WORK, real, dimension( * ) RWORK, integer, dimension( * )
    IWORK, integer, dimension( * ) IFAIL, integer INFO)

`CHPGVX` 🦋

Purpose:

CHPGVX computes selected eigenvalues and, optionally, eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and B are assumed to be Hermitian, stored in packed format, and B is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. not me getting to CHOOSE which eigenvalues I want, it's giving build-your-own-eigen-adventure 🎀

Parameters

`ITYPE`

ITYPE is INTEGER

Specifies the problem type to be solved:

= 1: A*x = (lambda)*B*x
= 2: A*B*x = (lambda)*x
= 3: B*A*x = (lambda)*x

`JOBZ`

JOBZ is CHARACTER*1

= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.

`RANGE`

RANGE is CHARACTER*1

= 'A': all eigenvalues will be found;
= 'V': all eigenvalues in the half-open interval (VL,VU] will be found;
= 'I': the IL-th through IU-th eigenvalues will be found.

Note the half-open interval: VL is not included but VU is. (VL,VU] — that's the vibe, don't forget the bracket situation bestie 🫶

`UPLO`

UPLO is CHARACTER*1

= 'U': Upper triangles of A and B are stored;
= 'L': Lower triangles of A and B are stored.

`N`

N is INTEGER

The order of the matrices A and B. N >= 0.

`AP`

AP is COMPLEX array, dimension (N*(N+1)/2)

AP 🦋

AP is COMPLEX array, dimension (N*(N+1)/2)

Okay so on entry, AP holds the upper or lower triangle of the Hermitian matrix A, packed columnwise into one linear array. The j-th column of A lives in AP like this:

if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

On exit, the contents of AP are destroyed. gone. reduced to atoms. so don't come looking for your original matrix in there bestie 😭

BP 💜

BP is COMPLEX array, dimension (N*(N+1)/2)

On entry, this is the upper or lower triangle of the Hermitian matrix B, packed columnwise in a linear array. The j-th column of B sits in BP like so:

if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.

On exit, BP holds the triangular factor U or L from the Cholesky factorization B = U**H*U or B = L*L**H, stored in the exact same format as B was. cute, consistent, iconic ✨

VL 🌸

VL is REAL

If RANGE='V', this is the lower bound of the interval to be searched for eigenvalues, and VL < VU. Not referenced if RANGE = 'A' or 'I'. so if you're not doing the interval thing, she's just sitting there unbothered 💅

VU 🌸

VU is REAL

If RANGE='V', this is the upper bound of the interval to be searched for eigenvalues, and again VL < VU. Not referenced if RANGE = 'A' or 'I'.

IL 🎀

IL is INTEGER

If RANGE='I', this is the index of the smallest eigenvalue to be returned. Constraint check: 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.

IU 🎀

IU is INTEGER

If RANGE='I', this is the index of the largest eigenvalue to be returned. Same rule as her sister: 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.

ABSTOL ⭐

ABSTOL is REAL

The absolute error tolerance for the eigenvalues, bestie. An approximate eigenvalue gets accepted as converged when it's determined to lie in an interval [a,b] whose width is less than or equal to:

        ABSTOL + EPS *   max( |a|,|b| ) ,

where EPS is the machine precision. If ABSTOL is less than or equal to zero, then EPS*|T| gets used in its place, where |T| is the 1-norm of the tridiagonal matrix you get by reducing AP to tridiagonal form.

Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*SLAMCH('S'), not zero. And if this routine comes back with INFO>0 — meaning some eigenvectors did not converge — try setting ABSTOL to 2*SLAMCH('S'). fr fr, this is the glow-up your convergence needs 🔥

M 🫶

M is INTEGER

The total number of eigenvalues found. Range: 0 <= M <= N. If RANGE = 'A', then M = N, and if RANGE = 'I', then M = IU-IL+1.

W 👑

W is REAL array, dimension (N)

On normal exit, the first M elements contain the selected eigenvalues in ascending order. sorted, organized, main character energy ✨

Z 🌈

Z is COMPLEX array, dimension (LDZ, N)

If JOBZ = 'N', then Z is not referenced, she's off the clock.

If JOBZ = 'V', then if INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i).

The eigenvectors are normalized as follows:

if ITYPE = 1 or 2, Z**H*B*Z = I;
if ITYPE = 3, Z**H*inv(B)*Z = I.

If an eigenvector fails to converge, then that column of Z contains the latest approximation to the eigenvector, and the index of that eigenvector is returned in IFAIL.

Note: the user must ensure that at least max(1,M) columns are supplied in the array Z. If RANGE = 'V', the exact value of M is not known in advance, so an upper bound must be used. no cap, plan ahead 🙈

LDZ 💫

LDZ is INTEGER

The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', then LDZ >= max(1,N).

WORK 🧚

WORK is COMPLEX array, dimension (2*N)

RWORK 🧚

RWORK is REAL array, dimension (7*N)

IWORK 🧚

IWORK is INTEGER array, dimension (5*N)

IFAIL 💀

IFAIL is INTEGER array, dimension (N)

INFO 🔥

INFO is INTEGER

= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value
> 0:  CPPTRF or CHPEVX returned an error code:
   <= N:  if INFO = i, CHPEVX failed to converge;
          i eigenvectors failed to converge.  Their indices
          are stored in array IFAIL.
   > N:   if INFO = N + i, for 1 <= i <= n, then the leading
          minor of order i of B is not positive definite.
          The factorization of B could not be completed and
          no eigenvalues or eigenvectors were computed.

so like: 0 is the slay, negative means you passed a bad argument at position i, and positive means the math said "not today" — either eigenvectors didn't converge (indices in IFAIL) or B isn't positive definite and literally nothing got computed. devastating, but at least it tells you 💅

Author 💖

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors: 🎀

Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

and that's a wrap on this whole entire saga bestie 🫶 generated automatically by Doxygen for LAPACK straight from the source code, sitting pretty at Version 3.11.0, dated Sun Nov 27 2022, and filed under complexOTHEReigen(3). mwah 💋✨

📦 where this page comes from

debian bookworm: liblapack-doc 3.11.0-2 · BSD-3-Clause
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash on 2026-10-14: a language model's rewrite of the original, under its terms

complexOTHEReigen(3)

✨ complexOTHEReigen(3) ✨

🌸 NAME 🌸

🦋 SYNOPSIS 🦋

🌈 Functions 🌈

🔥 Detailed Description 🔥

👑 Function Documentation 👑

💖 subroutine cggglm 💖

🌟 Purpose 🌟

🎀 Parameters 🎀

💫 Author 💫

💖 subroutine chbev 💖

🌟 Purpose 🌟

🎀 Parameters 🎀

Author

subroutine chbev_2stage

Purpose

Parameters

Author

Further Details

subroutine chbevd

Purpose

Parameters

🎀 Author

🦋 subroutine chbevd_2stage

✨ Purpose

🌸 Parameters

💖 Author

🔥 Further Details

🫶 subroutine chbevx

✨ Purpose

🌸 Parameters

👑 Author

🧚 subroutine chbevx_2stage

🎯 Purpose:

🌸 Parameters

👑 Author

🔍 Further Details:

🦋 CHBGV

🎯 Purpose:

🌸 Parameters

👑 Author

🌈 CHBGVD

🎯 Purpose:

✨ Parameters ✨

👑 Author 👑

🫶 Contributors: 🫶

🌸 subroutine chbgvx 🌸

🎯 Purpose 🎯

✨ Parameters ✨

IL 💜

IU ✨

ABSTOL 🎀

M 🌸

W 🦋

Z 💅

LDZ 🔥

WORK

RWORK

IWORK

IFAIL 👑

INFO 💀

Author

Contributors:

subroutine chpev ✨

Purpose:

Parameters

Author

subroutine chpevd 🌈

Purpose:

Parameters

🎀 Author 🎀

🦋 subroutine chpevx 🦋

✨ Purpose ✨

💖 Parameters 💖

🌸 Author 🌸

🌈 subroutine chpgv 🌈

✨ Purpose ✨

💅 Parameters 💅

W

subroutine `chbev_2stage`

subroutine `chbevd`

🦋 subroutine `chbevd_2stage`

🫶 subroutine `chbevx`

🧚 `subroutine chbevx_2stage`

`IL` 💜

`IU` ✨

`ABSTOL` 🎀

`M` 🌸

`W` 🦋

`Z` 💅

`LDZ` 🔥

`WORK`

`RWORK`

`IWORK`

`IFAIL` 👑

`INFO` 💀

`subroutine chpev` ✨

`subroutine chpevd` 🌈

🦋 subroutine `chpevx` 🦋

🌈 subroutine `chpgv` 🌈

`W`

`Z`

`LDZ`

`WORK`

`RWORK`

`INFO`

subroutine `chpgvd`

`CHPGVD` 💖

`ITYPE`

`JOBZ`

`UPLO`

`N`

`AP`

`BP`

`W`

`Z`

`LDZ`

`WORK`

`LWORK`

`RWORK`

`LRWORK`

`IWORK`

`LIWORK`

`INFO`

subroutine `chpgvx`

`CHPGVX` 🦋

`ITYPE`

`JOBZ`

`RANGE`

`UPLO`

`N`

`AP`